# Local convergence comparison between two novel sixth order methods for solving equations

## Keywords:

Jarratt-like method, sixth order of convergence, local convergence, Banach space, Frechet-derivative## Abstract

The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

## References

Argyros, Ioannis K., Santhosh George and Narayan Thapa. Mathematical modeling for the solution of equations and systems of equations with applications. Vol. 1. New York: Nova Publishes, 2018.

Argyros, Ioannis K., Santhosh George and Narayan Thapa. Mathematical modeling for the solution of equations and systems of equations with applications. Vol. 2. New York: Nova Publishes, 2018.

Argyros, Ioannis K., Munish Kansal and V. Kanwar, V. "Local convergence for multipoint methods using only the first derivative." SeMA J. 73, no. 4 (2016): 369-378.

Candela, V. and A. Marquina. "Recurrence relations for rational cubic methods. I. The Halley method." Computing 44, no. 2 (1990): 169-184.

Candela, V. and A. Marquina. "Recurrence relations for rational cubic methods. II. The Chebyshev method." Computing 45, no. 4 (1990): 355-367.

Chen, Jinhai, Ioannis Argyros and Ravi P. Agarwal. "Majorizing functions and two-point Newton-type methods." J. Comput. Appl. Math. 234, no. 5 (2010): 1473-1484.

Ezquerro, J.A., D. González and M.A. Hernández. "On the local convergence of Newton’s method under generalized conditions of Kantorovich." Appl. Math. Lett. 26, no. 5 (2013): 566-570.

Gutiérrez, J.M. and M.A. Hernández. "An acceleration of Newton’s method: super-Halley method." Appl. Math. Comput. 117, no. 2-3, (2001): 223-239.

Hernández, M.A. and N. Romero. "On a characterization of some Newton-like methods of R-order at least three." J. Comput. Appl. Math. 183, no. 1 (2005): 53-66.

Hernández-Verón, M.A. and Eulalia Martínez. "On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions." Numer. Algorithms 70, no. 2 (2015): 377-392.

Jarratt, P. "Some fourth order multipoint iterative methods for solving equations." Math. Comput. 20, no.95 (1966): 434-437.

Kantorovich, L.V. and G.P. Akilov. Functional analysis. Oxford-Elmsford, N.Y.: Pergamon Press, 1982.

Kumar, Abhimanyu and D.K. Gupta. "Local convergence of Super Halley’s method under weaker conditions on Fréchet derivative in Banach spaces." J. Anal. (2017) DOI:10.1007/s41478-017-0034-9.

Madhu, Kalyanasundaram. "Sixth order Newton-type method for solving system of nonlinear equations and its applications." Appl. Math. E-Notes 17 (2017): 221-230.

Parida, P.K. and D.K. Gupta. "Recurrence relations for a Newton-like method in Banach spaces." J. Comput. Appl. Math. 206, no. 2 (2007): 873-887.

Ren, Hongmin. "On the local convergence of a deformed Newton’s method under Argyros-type condition." J. Math. Anal. Appl. 321, no. 1 (2006): 396-404.

Rheinboldt, Werner C. "An adaptive continuation process for solving systems of nonlinear equations. vol. 3 of Banach Center Publ., 129-142.Warsaw: PWN, 1978.

Traub, J. F. Iterative methods for the solution of equations. Englewood Cliffs, N.J.: Prentice-Hall Series in Automatic Computation Prentice-Hall, Inc., 1964.

Wang, Xiuhua, Jisheng Kou and Chuanqing Gu. "Semilocal convergence of a sixthorder Jarratt method in Banach spaces." Numer. Algorithms 57, no. 4 (2011): 441-456.

Wu, Qingbiao and Yueqing Zhao. "Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space." Appl. Math. Comput. 175, no. 2 (2006): 1515-1524.

## Downloads

## Published

## How to Cite

*Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica*,

*18*, 5–19. Retrieved from https://studmath.up.krakow.pl/article/view/7635